The numbering of
references follows the order of the list of publications provided as
a separate document.

1
Investigating weak topologies for probability measures

My
early research was focused on weak topologies of probability measures
associated to stochastic processes, via Martingale Theory and the
General Theory of Processes. The first article I published in this
direction of research was [3] devoted to the convergence in
distribution of continuous martingales. This was followed and
improved by a series ([4], [5], [6]) of notes which included a first
version of a Central Limit Theorem for (discontinuous) square
integrable Martingales, building up a new method to approach the
convergence in distribution of stochastic processes. The first
applications of this method appeared in [7] and [8]. One of my main
papers on the method I built up is the memoir [9].

Furthermore,
several extensions and improvements of the method followed in
[10]–[14]. The article [15] provided a complete description of
tightness on the Skorokhod space D by means of stopping times.
On the other hand, in [16] the more general result on the Central
Limit Theorem for Local Martingales was established. My research
turned then into the search of necessary and sufficient conditions
for the validity of the Central Limit Theorem. This was achieved
for the case of semimartingales in [17].

Semimartingale
or martingale problems have been considered in [18] to [22] and [26].
In particular, various results on the approximation of diffusions
were derived. Finally, in [23], a paper written jointly with Eckhard
Platen, we analysed discretisation procedures and the approximation
of diffusions.

The
study of metastable phenomena in stochastic particle models,
motivated the search of a weaker topology on the space D
replacing the customary Skorokhod’s topology. The articles [24],
[25], [27], [28] were aimed at solving that problem.

Finally,
the convergence of non adapted processes was covered in [32], [33],
[36], and random field convergence was studied in [35].

2
An incursion in Stochastic Mechanics

I
came to Stochastic Mechanics in 1988, to work in collaboration with
physicists in Quantum Optics, (see references [29], [30], [31]). In
particular [31] introduced the concept of an entropic diffusion
which gives coherent grounds for building up Nelson’s approach to
Stochastic Mechanics. Several lecture notes in Spanish were published
in proceedings of various Chilean meetings. This
was a short passage through this theory. I arrived at the opinion
that Stochastic Mechanics is more of a simulation of Quantum
Mechanics based on classical stochastic processes. And the
probability space was depending on the observable chosen. Thus, to
overcome this difficulty, a new approach to probability was
necessary, and that was provided by non commutative approaches (or
Quantum Probability). That
is, Quantum Mechanics intrinsically contains a model for probability,
which extends the one proposed by Kolmogorov.In
my opinion, researchers in Probability need to be aware of both
models.

3
Non commutative stochastic analysis and open quantum systems

Applications
of Probability to Scattering Theory motivated a joint research with
Claudio Fernandez. Simultaneously, I became interested in non
commutative probability through the seminal work of Accardi,
Parthasarathy and Meyer. References [36] to [38], [40], [42],
and [45] to [47] are connected with the beginning of this direction
of research.

A
systematic study of Quantum Markov Semigroups started then. The focus
was, firstly, set on their qualitative analysis. An important part of
this research program was carried jointly with Franco Fagnola.
References [57] to [66], [71], [75], [77], [81], [82], are related to
this. Namely, those papers were aimed at answering some fundamental
questions like: Under which conditions on its generator there exists
an invariant state for a given quantum Markov semigroup? Is the
system ergodic? Is it recurrent or transient?

In
parallel, I started a research on the so called "quantum
decoherence" phenomenon, connected with classical reductions of
quantum Markov semigroups (see [67], [68], [69], [75], [80]).

Classical
reductions have recently been used too in designing statistical
inference on open quantum systems (see [84]).

On
the other hand, classical dilations of quantum Markov semigroups have
been investigated in a number of papers (see [80] as well as [72],
[74] both written jointly with Carlos Mora).

Even
though the paradigm of Markov approximation to open quantum systems
still having a number of relevant questions to be solved, it is
currently important for applications to consider non Markov
approaches. This is part of a research started in 2007 with Andrzej
Kossakowski (see [73], [76], [78], [79]).

4
Applications to physics, engineering and finance

Throughout
my career as a research fellow in Mathematics I have been very often
inspired by physical problems. Perhaps Quantum Optics has been the
most influential field in my latest research as it follows from the
papers [30], [31], [43], [47], [52]. However, engineering
applications are not missing: in [50] we applied Stochastic
Differential Equations to Electricity. Moreover, in a series of
papers with Eckhard Platen (see e.g. [44], [46], [49]), applications
of Stochastic Analysis to Finance have been developed.